1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
|
SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
* ..
*
* Purpose
* =======
*
* DSPTRD reduces a real symmetric matrix A stored in packed form to
* symmetric tridiagonal form T by an orthogonal similarity
* transformation: Q**T * A * Q = T.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
* On entry, the upper or lower triangle of the symmetric matrix
* A, packed columnwise in a linear array. The j-th column of A
* is stored in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
* if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
* On exit, if UPLO = 'U', the diagonal and first superdiagonal
* of A are overwritten by the corresponding elements of the
* tridiagonal matrix T, and the elements above the first
* superdiagonal, with the array TAU, represent the orthogonal
* matrix Q as a product of elementary reflectors; if UPLO
* = 'L', the diagonal and first subdiagonal of A are over-
* written by the corresponding elements of the tridiagonal
* matrix T, and the elements below the first subdiagonal, with
* the array TAU, represent the orthogonal matrix Q as a product
* of elementary reflectors. See Further Details.
*
* D (output) DOUBLE PRECISION array, dimension (N)
* The diagonal elements of the tridiagonal matrix T:
* D(i) = A(i,i).
*
* E (output) DOUBLE PRECISION array, dimension (N-1)
* The off-diagonal elements of the tridiagonal matrix T:
* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*
* TAU (output) DOUBLE PRECISION array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* If UPLO = 'U', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(n-1) . . . H(2) H(1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
* overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
*
* If UPLO = 'L', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(1) H(2) . . . H(n-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a real scalar, and v is a real vector with
* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
* overwriting A(i+2:n,i), and tau is stored in TAU(i).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, HALF
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
$ HALF = 1.0D0 / 2.0D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, I1, I1I1, II
DOUBLE PRECISION ALPHA, TAUI
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPTRD', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* I1 is the index in AP of A(1,I+1).
*
I1 = N*( N-1 ) / 2 + 1
DO 10 I = N - 1, 1, -1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(1:i-1,i+1)
*
CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
E( I ) = AP( I1+I-1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(1:i,1:i)
*
AP( I1+I-1 ) = ONE
*
* Compute y := tau * A * v storing y in TAU(1:i)
*
CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
$ 1 )
*
* Compute w := y - 1/2 * tau * (y'*v) * v
*
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
*
AP( I1+I-1 ) = E( I )
END IF
D( I+1 ) = AP( I1+I )
TAU( I ) = TAUI
I1 = I1 - I
10 CONTINUE
D( 1 ) = AP( 1 )
ELSE
*
* Reduce the lower triangle of A. II is the index in AP of
* A(i,i) and I1I1 is the index of A(i+1,i+1).
*
II = 1
DO 20 I = 1, N - 1
I1I1 = II + N - I + 1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(i+2:n,i)
*
CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
E( I ) = AP( II+1 )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to A(i+1:n,i+1:n)
*
AP( II+1 ) = ONE
*
* Compute y := tau * A * v storing y in TAU(i:n-1)
*
CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
$ ZERO, TAU( I ), 1 )
*
* Compute w := y - 1/2 * tau * (y'*v) * v
*
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
$ 1 )
CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
$ AP( I1I1 ) )
*
AP( II+1 ) = E( I )
END IF
D( I ) = AP( II )
TAU( I ) = TAUI
II = I1I1
20 CONTINUE
D( N ) = AP( II )
END IF
*
RETURN
*
* End of DSPTRD
*
END
|
